OFFSET
1,3
COMMENTS
T(n,k) = T(n,k-1) for k>A186971(n). The triangle contains all values of T up to the last element of each row that is different from its predecessor.
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
FORMULA
T(n,k) = Sum_{i=1..k} A186972(n,i).
EXAMPLE
T(5,3) = 10 because there are 10 subsets of {1,2,3,4,5} containing n and having <=3 pairwise coprime elements: {5}, {1,5}, {2,5}, {3,5}, {4,5}, {1,2,5}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}.
Triangle T(n,k) begins:
1;
1, 2;
1, 3, 4;
1, 3, 4;
1, 5, 10, 12;
1, 3, 4;
1, 7, 18, 26, 28;
MAPLE
with(numtheory):
s:= proc(m, r) option remember; mul(`if`(i<r, i, 1), i=factorset(m)) end:
a:= proc(n) option remember; `if`(n<4, n, pi(n)-nops(factorset(n))+2) end:
b:= proc(t, n, k) option remember; local c, d, h;
if k=0 or k>n then 0
elif k=1 then 1
elif k=2 and t=n then `if`(n<2, 0, phi(n))
else c:= 0;
d:= 2-irem(t, 2);
for h from 1 to n-1 by d do
if igcd(t, h)=1 then c:= c +b(s(t*h, h), h, k-1) fi
od; c
fi
end:
T:= proc(n, k) option remember;
b(s(n, n), n, k) +`if`(k=0, 0, T(n, k-1))
end:
seq(seq(T(n, k), k=1..a(n)), n=1..20);
MATHEMATICA
s[m_, r_] := s[m, r] = Product[If[i < r, i, 1], {i, FactorInteger[m][[All, 1]]}]; a[n_] := a[n] = If[n < 4, n, PrimePi[n]-Length[FactorInteger[n]]+2]; b[t_, n_, k_] := b[t, n, k] = Module[{c, d, h}, Which[k == 0 || k > n, 0, k == 1, 1, k == 2 && t == n, If[n < 2, 0, EulerPhi[n]], True, c = 0; d = 2-Mod[t, 2]; For[h = 1, h <= n-1, h = h+d, If[GCD[t, h] == 1, c = c+b[s[t*h, h], h, k-1] ] ]; c ] ]; t[n_, k_] := t[n, k] = b[s[n, n], n, k]+If[k == 0, 0, t[n, k-1]]; Table[Table[t[n, k], {k, 1, a[n]}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)
CROSSREFS
KEYWORD
AUTHOR
Alois P. Heinz, Mar 02 2011
STATUS
approved